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arxiv: 2605.15668 · v1 · pith:BB6YFE4Enew · submitted 2026-05-15 · 🧮 math.OC

Viscosity-Informed Generative Actor-Critic for High-Dimensional Stochastic Optimal Control

Alen E. Golpashin , Gokul Puthumanaillam , Melkior Ornik , Bruce A. Conway This is my paper

Pith reviewed 2026-05-20 17:35 UTC · model grok-4.3

classification 🧮 math.OC
keywords viscosity solutionsHamilton-Jacobi-Bellman equationsstochastic optimal controlactor-critic methodsmin-max optimizationdegenerate elliptic PDEsexit-time problemsgenerative models
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The pith

Under structural assumptions, uniform limit points of generative actor-critic value approximations satisfy viscosity inequalities for degenerate elliptic HJB equations on sampled test families.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a generative actor-critic algorithm that approximates viscosity solutions to stationary Hamilton-Jacobi-Bellman equations arising in stochastic exit-time control on bounded domains. Viscosity enforcement is cast as a min-max optimization over an envelope-generated family of test functions, each parameterized by a symmetric positive definite matrix. Under the stated structural and asymptotic conditions on the dynamics, cost, and approximation sequence, every uniform limit point of the learned value functions satisfies the viscosity inequalities with respect to that test family. Numerical tests indicate that the resulting policies exhibit fewer empirical viscosity violations and retain performance when the underlying dynamics are perturbed.

Core claim

Any uniform limit point of the sequence of value-function approximations produced by the viscosity-informed generative actor-critic satisfies the viscosity inequalities on the sampled envelope-generated test family, thereby furnishing a viscosity solution to the target stationary degenerate elliptic HJB equation.

What carries the argument

Min-max formulation of viscosity enforcement over an envelope-generated test family parameterized by symmetric positive definite matrices.

If this is right

  • The learned policies can be used directly for high-dimensional stochastic exit-time problems without post-hoc projection onto viscosity solutions.
  • Empirical viscosity violations are reduced relative to standard actor-critic baselines on the same HJB problems.
  • Performance remains stable when the true dynamics are replaced by perturbed versions inside the same structural class.
  • The approach scales to state dimensions where classical grid-based viscosity solvers become intractable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same min-max envelope construction could be ported to time-dependent or infinite-horizon problems whose HJB equations are still degenerate elliptic.
  • If the test-family sampling density is increased with dimension, the method might serve as a practical surrogate for checking uniqueness of viscosity solutions in high-dimensional control.
  • Combining the learned value function with classical verification theorems would immediately yield optimality certificates for the extracted feedback policies.

Load-bearing premise

The dynamics, running cost, and sequence of approximations must obey the structural and asymptotic conditions that allow uniform limit points to inherit the viscosity inequalities from the min-max formulation.

What would settle it

Construct a concrete dynamics-cost pair satisfying the structural assumptions, run the algorithm to produce a sequence of value approximations, extract a uniform limit point, and exhibit a test function from the sampled family for which the viscosity inequality fails at some interior point.

read the original abstract

We introduce a method for approximating viscosity solutions of stationary degenerate elliptic Hamilton--Jacobi--Bellman equations on bounded domains arising in stochastic exit-time control. Viscosity enforcement is formulated as a min--max problem over an envelope-generated test family parameterized by symmetric positive definite matrices. Under structural and asymptotic assumptions, any uniform limit point of the value function approximations satisfies the viscosity inequalities on the sampled test family. Numerical experiments show that the proposed method reduces empirical viscosity violations and improves robustness under perturbed dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a viscosity-informed generative actor-critic method for approximating viscosity solutions of stationary degenerate elliptic Hamilton-Jacobi-Bellman equations arising in high-dimensional stochastic exit-time control on bounded domains. Viscosity enforcement is formulated as a min-max problem over an envelope-generated test family parameterized by symmetric positive definite matrices. Under structural and asymptotic assumptions on the dynamics, cost, and approximation sequence, any uniform limit point of the value function approximations is shown to satisfy the viscosity inequalities on the sampled test family. Numerical experiments indicate that the proposed method reduces empirical viscosity violations and improves robustness under perturbed dynamics.

Significance. If the convergence result holds under the stated assumptions, the work provides a structured approach to incorporating viscosity solution concepts into actor-critic methods for high-dimensional stochastic control, where classical solutions may fail to exist. The min-max formulation over the envelope-generated test family represents a novel mechanism for enforcing viscosity properties in a generative setting and could extend to other degenerate elliptic problems if the assumptions prove verifiable in applications.

major comments (1)
  1. [Theoretical section presenting the main convergence result] The central theoretical claim (connecting the min-max formulation to viscosity inequalities for uniform limit points) is explicitly conditional on structural and asymptotic assumptions regarding the dynamics, cost, and approximation sequence. These assumptions must be stated explicitly with equation or proposition references in the main theoretical section, as their precise form determines the scope and applicability of the result.
minor comments (2)
  1. [Numerical experiments section] The numerical experiments report reductions in empirical viscosity violations but provide no quantitative error bars, standard deviations, or details on the sampling procedure for the test family and the specific form of the perturbed dynamics; adding these would strengthen the empirical support without altering the central claim.
  2. [Method formulation] Clarify the parameterization of the test family by symmetric positive definite matrices and any related notation in the min-max objective for improved readability and reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

  1. Referee: [Theoretical section presenting the main convergence result] The central theoretical claim (connecting the min-max formulation to viscosity inequalities for uniform limit points) is explicitly conditional on structural and asymptotic assumptions regarding the dynamics, cost, and approximation sequence. These assumptions must be stated explicitly with equation or proposition references in the main theoretical section, as their precise form determines the scope and applicability of the result.

    Authors: We agree that the convergence result is conditional on these assumptions and that their explicit statement with equation or proposition references in the main theoretical section will improve clarity and help readers assess the scope of the result. In the revised manuscript we will insert a dedicated paragraph (or short subsection) immediately preceding the statement of the main convergence theorem that enumerates all structural and asymptotic assumptions, each accompanied by its precise equation or proposition reference from the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper claims that under structural and asymptotic assumptions on the dynamics, cost, and approximation sequence, any uniform limit point of the value function approximations satisfies the viscosity inequalities on the sampled test family. This is presented as following from standard viscosity solution techniques for degenerate elliptic HJB equations applied to the min-max formulation over the envelope-generated test family. No step in the provided abstract or description reduces a prediction or central result to a fitted parameter, self-definition, or self-citation chain by construction. The numerical experiments are described as empirical support for reduced violations rather than a substitute for the conditional theorem. The derivation is self-contained against external benchmarks once the stated assumptions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background results from viscosity solution theory for degenerate elliptic HJB equations together with problem-specific structural and asymptotic assumptions whose precise content is not expanded in the abstract.

axioms (1)
  • domain assumption Structural and asymptotic assumptions on the dynamics, running cost, and approximation sequence
    Invoked in the abstract to guarantee that uniform limit points of the learned value functions satisfy the viscosity inequalities on the sampled test family.

pith-pipeline@v0.9.0 · 5618 in / 1230 out tokens · 99587 ms · 2026-05-20T17:35:26.673240+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Viscosity enforcement is formulated as a min–max problem over an envelope-generated test family parameterized by symmetric positive definite matrices. Under structural and asymptotic assumptions, any uniform limit point of the value function approximations satisfies the viscosity inequalities on the sampled test family.

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We introduce the envelope construction and the associated envelope-jet penalty... J(V) := E_{(x,M)} [J_super + J_sub]

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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